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A334980
a(n) is the total number of down steps between the (n-1)-th and n-th up steps in all 3_2-Dyck paths of length 4*n. A 3_2-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -2.
5
0, 3, 31, 248, 1941, 15334, 122915, 999456, 8231740, 68562887, 576661761, 4891506968, 41801697070, 359574305580, 3111012673755, 27055673506128, 236387476114548, 2073957836402524, 18264689865840284, 161403223665821280, 1430768729986730685, 12719497076318052990
OFFSET
0,2
COMMENTS
For n = 1, there is no (n-1)-th up step, a(1) = 3 is the total number of down steps before the first up step.
LINKS
A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.
FORMULA
a(0) = 0 and a(n) = 3*binomial(4*n+7, n+1)/(4*n+7) - 12*binomial(4*n+3, n)/(4*n+3) for n > 0.
EXAMPLE
For n = 2, the 3_2-Dyck paths are UDDDDDUD, UDDDDUDD, UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD, DUDDDDUD, DUDDDUDD, DUDDUDDD, DUDUDDDD, DUUDDDDD, DDUDDDUD, DDUDDUDD, DDUDUDDD, DDUUDDDD. Therefore the total number of down steps between the first and second up steps is a(2) = 5 + 4 + 3 + 2 + 1 + 0 + 4 + 3 + 2 + 1 + 0 + 3 + 2 + 1 + 0 = 31.
MATHEMATICA
a[0] = 0; a[n_] := 3*Binomial[4*n+7, n+1]/(4*n + 7) - 12 * Binomial[4*n + 3, n]/(4*n + 3); Array[a, 22, 0]
PROG
(SageMath) [3*binomial(4*n + 7, n + 1)/(4*n + 7) - 12*binomial(4*n + 3, n)/(4*n + 3) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 19 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Sarah Selkirk, May 18 2020
STATUS
approved